Novel Product Manifold Modeling and Orthogonality-Constrained Neural Network Solver for Parameterized Generalized Inverse Eigenvalue Problems

TL;DR

This paper introduces a novel neural network approach on product manifolds to solve parameterized generalized inverse eigenvalue problems, enabling end-to-end training with improved optimization capabilities.

Contribution

It develops a new model on product manifolds and proposes a parameterized orthogonality-constrained neural network for PGIEP, advancing optimization techniques in this domain.

Findings

Effective solution demonstrated through numerical experiments

Gradient Lipschitz continuity of the objective function proved

End-to-end training without alternating optimization achieved

Abstract

A parameterized orthogonality-constrained neural network is proposed for the first time to solve the parameterized generalized inverse eigenvalue problem (PGIEP) on product manifolds, offering a new perspective to address PGIEP. The key contributions are twofold. First, we construct a novel model for the PGIEP, where the optimization variables are located on the product of a Stiefel manifold and a Euclidean manifold. This model enables the application of optimization algorithms on the Stiefel manifold, a capability that is not achievable with existing models. Additionally, the gradient Lipschitz continuity of the objective function is proved. Second, a parameterized Stiefel multilayer perceptron (P-SMLP) that incorporates orthogonality constraints is proposed. Through hard constraints, P-SMLP enables end-to-end training without the need of alternating training between the two manifolds, providing a robust computational framework for generic PGIEPs. Numerical experiments demonstrate the effectiveness of the proposed method.